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Definability of the natural numbers in totally real towers of nested square roots

Authors: Xavier Vidaux and Carlos R. Videla
Journal: Proc. Amer. Math. Soc. 143 (2015), 4463-4477
MSC (2010): Primary 03B25; Secondary 11U05, 11R80
Published electronically: March 18, 2015
MathSciNet review: 3373945
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Abstract: For the ring of integers $ \mathcal {O}$ of a totally real algebraic field, Julia Robinson defines a set $ A(\mathcal {O})$ such that either $ A(\mathcal {O})=\{+\infty \}$ or it is an interval in $ \mathbb{R}$. She then proves that if this set has a minimum, then the natural numbers can be defined in $ \mathcal {O}$, and hence $ \mathcal {O}$ has undecidable first-order theory. All known examples are such that $ A(\mathcal {O})$ has a minimum which is either $ 4$ or $ +\infty $. In this work, we construct two infinite families of subrings of such rings for which $ \inf (A(\mathcal {O}))$ is strictly between $ 4$ and $ +\infty $. In one family, the infimum is a minimum, whereas in the other family it is not, but we can still define the natural numbers in this case.

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Additional Information

Xavier Vidaux
Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160 C, Concepción, Chile

Carlos R. Videla
Affiliation: Department of Mathematics, Physics and Engineering, Mount Royal University, 4825 Mount Royal Gate SW, Calgary, Alberta, Canada T3E 6K6

Received by editor(s): November 26, 2013
Received by editor(s) in revised form: June 28, 2014
Published electronically: March 18, 2015
Additional Notes: Both authors have been supported by the first author’s Chilean research projects Fondecyt 1090233 and 1130134, and by Mount Royal University PD Funds.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

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